Slant Geometry

Epstein [1986] introduced the hyperbolic Gauss map of a surface in the Poincare ball model of the hyperbolic plane. The definition is given in an intrinsic way which makes it hard to study the singularities of that Gauss map. In one of the most important papers in applications of singularity theory to differential geometry, S. Izumiya and his co-authors [2003] defined the hyperbolic Gauss map of a hypersurface in the Minkowski space model of the hyperbolic space. The work set the foundations of applications of singularity theory to the extrinsic geometry of submanifolds in semi-Euclidean spaces, which is where this proposal belongs. In [Izumiya etal, 2003] are constructed two Gauss maps of a hypersurface in the Minkowski space model of the hyperbolic space: the hyperbolic Gauss map and the de Sitter Gauss map. These are related to the contact of the hypersurface with, respectively, hyperplanes and horospheres. The aim of the project is to construct a 1-parameter family of Gauss maps that join the hyperbolic and the de Sitter Gauss maps, providing thus a link between the hyperbolic geometry and the horospherical geometry.

The research will have direct impact on the PI, his collaborators and his PhD students, people working in singularity theory in the UK and worldwide, and undergraduate students at Durham University. (See Impact Plan for more details.)